Let $\phi_{\alpha}(z)=\frac{z-\alpha}{1-\bar{\alpha}z}$ for $0<|\alpha|<1$
Find all the line $L$ in the complex plane such that $\phi_{\alpha} (L)=L$
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2026-03-28 03:00:52.1774666852
Möbius transformation of the complex plane
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Hint: Möbius transformations take lines and circles to lines and circles. The lines are distinguished from circles in that they go through $\infty$ (in the extended complex plane, i.e. the Riemann sphere). What does this transformation do to $\infty$, and what does it map to $\infty$?